This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 132.64.33.204

This content was downloaded on 13/09/2014 at 17:24

Please note that terms and conditions apply.

The multilevel four-stroke swap engine and its environment

View the table of contents for this issue, or go to the journal homepage for more

2014 New J. Phys. 16 095003

(http://iopscience.iop.org/1367-2630/16/9/095003)

Home Search Collections Journals About Contact us My IOPscience

The multilevel four-stroke swap engine and itsenvironment

Raam Uzdin and Ronnie KosloffFritz Haber Research Center for Molecular Dynamics, Hebrew University of Jerusalem,Jerusalem 91904, IsraelE-mail: raam@mail.huji.ac.il and ronnie@fh.huji.ac.il

Received 19 May 2014, revised 9 July 2014Accepted for publication 15 July 2014Published 12 September 2014

New Journal of Physics 16 (2014) 095003

doi:10.1088/1367-2630/16/9/095003

AbstractA multilevel four-stroke engine where the thermalization strokes are generatedby unitary collisions with thermal bath particles is analyzed. Our model issolvable even when the engine operates far from thermal equilibrium and in thestrong system–bath coupling. Necessary operation conditions for the heatmachine to perform as an engine or a refrigerator are derived. We relate the workand efficiency of the device to local and non-local statistical properties of thebaths (purity, index of coincidence, etc) and put upper bounds on these quan-tities. Finally, in the ultra-hot regime, we analytically optimize the work and finda striking similarity to results obtained for efficiency at maximal power ofclassical engines. The complete swap limit of our results holds for any four-stroke quantum Otto engine that is coupled to the baths for periods that aresignificantly longer than the thermal relaxation time.

Keywords: quantum thermodynamics, heat engine, multilevel, Otto engine

1. Introduction

Present day technology is on the verge of enabling different realizations of quantum heatmachines where the engine core (‘working substance’) comprises of a single particle in adiscrete level system (e.g. a qubit). In analogy to their classical counterparts, quantum heatmachines can be used to cool or to produce work. Furthermore, similarly to classical

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal

citation and DOI.

New Journal of Physics 16 (2014) 0950031367-2630/14/095003+26$33.00 © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

thermodynamics, performance analysis of heat machines without going into the details of aspecific realization is a key theme in quantum thermodynamics (QT). There are severalapproaches to QT. The more recent one comes from quantum information resource theory. Inthis approach thermal states are considered free while non-equilibrium states are considered as aresource ([1–3] and references therein). An energy conserving unitary is used to couple thesystem and the bath. A different viewpoint to QT is dynamical, and uses the framework ofquantum generators of open systems [4]. In this approach, models of a quantum heat machineare constructed and analyzed. The methodology in this study overlaps with both approaches. Onthe one hand, an explicit dynamical analysis of an engine is carried out, and on the otherhand the thermalization process is based on energy-conserving unitary operation betweenthe system and bath. This is different from studies in the dynamical approach where theinteraction with the bath is modeled by some effective reduced-dynamics non-unitary evolution(Lindblad equation). A third approach called typicality aims to find the conditions whencomplex interactions lead to thermal behavior for a typical set of states (see [5] and referencestherein).

A useful theory of QT should provide simple and efficient tools to evaluate theperformance of quantum heat machines. However, as we show here, in a multilevel system, it isfar from trivial to even map the parameter regimes where the machine operates as an engine oras a refrigerator (let alone their performance). The goal of this work is to gain a morecomprehensive understanding of the operation of multilevel quantum heat machines and tostudy the regimes in which the machines operate as an engine or as a refrigerator. In particular,in our model the dynamics can be solved analytically even when the coupling between the bathand the system is strong (this is a very difficult and subtle limit in the standard open systemsapproach [4]). Our analysis emphasizes the difference between ‘non-local’ quantities thatcannot be evaluated using a single bath, and ‘local’ quantities that can be evaluated using asingle bath. In principle, we wish to express non-local quantities of interest such as work, heatand efficiency in terms of simpler local quantities (temperature, purity of the baths, entropy ofthe baths, etc).

Various quantum engine models have been suggested—some are continuous (where thebaths are always connected to the engine) and some are reciprocating (where in different strokesdifferent processes take place). The equivalence between a laser and the Carnot engine [6]has inspired the study of quantum heat engines. For example, studying maximumpower operation [7–10], the role of coherence and entanglement [11–17], quantum refrigeratorsand the third law of thermodynamics [18–25], and the connection to quantum information[26, 27]. Recently, machines powered by non-thermal baths have been studied as well [28–30].

Different quantum working mediums have been considered: two-level systems, N-levelsystems or harmonic oscillators [31–37]. In most previous studies the system bath dynamicswas modeled by a reduced description where the dynamics of the system is described explicitlyby a master equation with a generator cast into the Lindblad form [38, 39]. In the reducedformalism the bath is only considered implicitly, therefore the effect of the engine on the bathsis almost always ignored. In this paper we take the opposite point of view and study the heatmachine operation using the properties of the baths. One of the main difficulties in the Lindbladapproach is that the reduced dynamics generators are not uniquely defined. Various choices ofthe Lindblad generator lead to the same reduced dynamics of the working medium but todifferent entropy fluxes to and from the baths. As it turns out, the Lindblad operators have to becarefully chosen in order to be consistent with the second law of thermodynamics [4].

2

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

The problem becomes more severe when time dependent external fields drive thesystem [40, 41].

The analysis carried out in this paper overcomes this problem by considering a very simpleyet physically plausible bath model that is based on collisions [42, 43]. In this model the bathconsists of non-interacting particles which are initially in the same thermal state. The engineinteracts with the bath particles via a two-particle collision described by a unitary operation. Wechoose to model the collision by the unitary swap operation but other unitary operations such asCNOT and random unitaries have been considered as well [44, 45] even though they were notapplied to the study of heat machines. As will be discussed later on, the swap operation has twointeresting physical limits. The first is the full swap that describes full thermalization and theother is the weak partial swap that mimics quasi-static evolution where the heat is equal to thebath temperature multiplied by the entropy difference.

Typically, in these collisions based baths the system that interacts with the bath is taken tobe a single qubit but qudits and a chain of coupled qubits have been considered as well [46, 47].The collision model can also represent a general non-Markovian dynamics [48, 49]. If,however, the bath is very large, a particle that interacted with the system is unlikely to affect thesystem again. In such cases the dynamics is Markovian. See [50, 51] and reference therein formore studies on collision-generated Markovian dynamics. In this article we consider Markoviancollisions in order to study the salient features of an engine driven by collision baths.

The importance of studying multilevel heat machines is twofold. First, experimentallyspeaking, in some systems it is not possible or not practical to interact with only two levels.Secondly, two-level systems may contain some non-generic features and it is highly importantto isolate the more general features of quantum heat machines by considering multilevelengines. For example, in a two-level system temperature is always well defined if the densitymatrix is diagonal in the energy basis. Clearly this is not true if there are more than two levels.

While in a two-level engine it is fairly simple to map the parameter space where the systemoperates as an engine and as a refrigerator, when the number of levels is three or more it is aconsiderably more complicated task [37]. In a two-level system the different operation regimesare determined by the temperature and the gap energy associated with the cold and hot strokes.It is not straightforward what energy scale replaces the two-level energy gap in the multilevelcase (the variance? the maximal gap?). Furthermore, it is not clear at this point if necessary andsufficient conditions for engine (or refrigerator) operation can be formulated without explicitlysolving the full dynamics of the system. In this work we provide the necessary conditions forthe operation of a multilevel Otto engine as engine or a refrigerator. In particular, in the ultra-hot regime we obtain a necessary and sufficient condition. Another important issue addressed inour analysis is work output optimization. In the ultra-hot regime we find the optimal way tocompress (or expand) a generic multilevel working substance in order to produce maximal workoutput per cycle.

This study can be related to ‘finite time thermodynamics’ [52–54] although there is noexplicit reference to time. Implicitly the cycle can be related to the collision timescale and inaddition there is no assumption of thermal equilibration of the device with the baths uponcontact.

The article is organized as follows: section 2 contains the description of our engine andbath model. In section 3 we analyze the steady state operation of the engine and discuss thedifference between quantum swap and classical random swap. Next, section 4 explores variousaspects of the engine evolution: Clausius and, generalized Clausius inequality and the bathsʼ

3

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

purity reduction. Necessary conditions for operation of a multilevel swap engine as an engine orrefrigerator are derived in section 5. In section 6 we set upper bounds on the work andefficiency in terms of statistical quantities like purity, index of coincidence and Woottersdistance. Finally, section 7 presents two possible regimes of operation: the ultra-hot bath regimeand the quasi-static regime.

2. The baths, engine, and their interaction

2.1. The bath

Our bath model is based on the ‘quantum homogenizer’ introduced in [42, 43, 55]. This bathcontains a very large amount of single species non-interacting multilevel particles. At thebeginning all the particles are in the same thermal state. However, each time a bath particleinteracts with the engine the population of the particle and of the engine changes according tothe simple partial swap rule that will be described shortly. The special case of a completepopulation swap was used in [33, 56, 57] to generate a thermal distribution in a finite time. Forother studies of reduced dynamics using unitary interaction with ancillary systems see [58–60].

Every time the system (engine) interacts with the bath it collides with a new thermalparticle in the bath. If the system is coupled only to one bath then repeated collision would leadto thermalization of the engine particle at an exponential rate. In this work, however, we willassume that there is only one collision or none at all at each thermal stroke of the engine.Therefore, the engine will typically not be in a Gibbs state. The extension to multiple collisionsin each thermal stroke is straightforward.

The bath is assumed to be large enough so that the probability to re-collide with a bathparticle that already interacted with the system is negligible. Consequently, the resultingdynamics is Markovian. We assume that the particles do not interact with each other in the bathbut only with the system particle at the bath–system interface. That is, in our model there is nothermalization inside the bath. In principle, for large baths, this has no impact on the engineʼsoperation. If the bath is sufficiently large it does not matter if the scattered particle getsthermalized again or not as the engine will (almost) always interact with a new thermal particle.Nevertheless, it is interesting to explore the operation of small baths with and without intra-baththermalization.

There is a different class of quantum engines where the baths are always connected to theengine but different levels are connected to different baths [18]. This implies that if only onebath is connected the relative probabilities of the specific coupled levels will be thermal but theengine as a whole will not be in a Gibbs state. Alternatively stated, these bath models have acontinuum of steady states. In our model the Gibbs state is the only single bath steady state.

2.2. The engineʼs cycle

Our engine comprises a multilevel system that is driven to a four-stroke Otto cycle [61].Figure 1(a) illustrates the engineʼs operation for a two-level system. In stroke A the gap spacingis increased by applying an external field. The engine is decoupled from the baths at this stage.The state evolution is such that the systemʼs density is diagonal in the energy basis at thebeginning and at the end of the stroke. Adiabatic change of the Hamiltonian will achieve such a

4

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

transformation but other faster options exist (see section 2.5). Despite the existence of fasteralternatives, for simplicity, we refer to this part of the cycle as an ‘adiabatic stroke’.

In stroke B the gap is kept fixed in time, but the engine is allowed to interact with the hotbath. During this time interval there may be one collision, multiple collisions or none at all. Theaverage collision rate is a parameter that characterizes the bath and its coupling to the engine.It encapsulates within the particle density in the bath, the engine cross-section for collision,etc. The collision entangles the engine and the particles in the bath. However since theinteracting bath particles will not interact with the engine again (the top spheres in figure 1(a)with temperature denoted by prime), we consider the reduced dynamics of the engine bytracing out the scattered particle of the bath. This stage changes the entropy of the engine andinvolves heat exchange with the hot bath. In stroke C the external field is changed again sothat the gap returns to its initial value. Finally, in stroke D, the system is coupled to thecold bath.

In the multilevel case the energy levelsʼ evolution can be considerably more intricate. First,we do not assume that all the levels are increased or decreased by the same ratio (figure 1(c)).Second, we allow the levels to cross (figure 1(d)). The engine scheme described here requiresexternal time-dependent control of the energy level. An autonomous implementation of thisengine that requires no control at all will be described elsewhere.

2.3. The collision model

We assume that at the initial state, the system and the bath particles are in a product of a thermalGibbs state:

Figure 1. (a) The energy of the ground state and excited state of the two level systemthat comprises a four-stroke, two-level, partial swap collision driven engine. (b)Dynamics of the ground state population. In strokes A and B the population is fixed andthe energy gap changes. In strokes C and D the gap is fixed and the population changesvia collisions with the bathʼs particles. In general the collision stage is not an isotherm.The work is done during strokes A and C, and heat exchange takes place during strokesB and D. (c) In a multilevel system the levelʼs dynamics can be more intricate. Thelevels do not have to be compressed by the same factor. Furthermore they can evencross each other (d) as long as it does not conflict with the energy population invariancein stages B and C.

5

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

ρ ρ ρ ρ= ⊗ ⊗ ...., (1)b b b

∑ρ = =−

=

− −−

e e e , (2)b ii

E

T

j

N E

T

E F

T,1

b i

b

b i

b

b i b

b

, , ,

ρ =≠ 0, (3)b i j,

where ‘b’ stands for ‘c’ (cold bath) or ‘h’ (hot bath). The free energy Fb takes care ofnormalization. Let U be some two-particle unitary operation that conserves the total energy ofthe two particles. The reduced density matrices after the interaction are:

ρ ρ ρ= ⊗′ †( )U Utr , (4)b s b s

ρ ρ ρ= ⊗′ †( )U Utr . (5)s b b s

When the collision Hamiltonian that generates U is invariant under the transformation ↔b sone can show that:

ρ ρ ρ ρ− = − −′ ′( ). (6)b b s s

This condition, together with the total energy conservation, implies that the energy levels of thebath and the system have to be equal.

2.4. Density matrix swap and energy population swap

In this work we focus on collisions that induce a single parameter convex transformation for thesystemʼs reduced density matrix or for the systemʼs energy population. The density swap ruleis:

ρ ρ ρ= − +′ x x(1 ) , (7)s s b

ρ ρ ρ= − +′ x x(1 ) , (8)b b s

⩽ ⩽x0 1, (9)

where x is the swap parameter. For x = 0 the density matrices remain as they were, and for x = 1the density matrix of the bath and the particle completely interchange. It is easy to verify thatthis transformation satisfies (6), and that the only steady state of this transformation is ρ ρ=b s.

Re-colliding the system particles with new bath particles in a thermal state will lead toexponential decay of ρs towards ρb. Yet, in this work, we do not assume that equilibrium isreached. In fact we will mostly consider a single collision at most in a given stroke (B or D).

Although the density swap rule has a very simple structure, our framework is valid for amore general type of swap operation. Let pb and ps denote the energy level populations of thebath and system particles. We shall use bold face characters to denote quantities that have alevel index such as the probability or energies of a bath Eb. The energy population swap rule is:

= − +′ x xp p p(1 ) , (10)s s b

= − +′ x xp p p(1 ) , (11)b b s

6

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

⩽ ⩽x0 1, (12)

where as before the prime relates to the traced-out outcome after the collision. We do not claimthat the swap operation in its current form is general enough to describe many typical thermalbaths used in experiments. Yet, such bath could in principle be built. One advantage of this bathmodel is that it leads to a solvable dynamics that can be used as a reference for other morecomplex models. Another advantage, as will be shown later on, is that it helps to formulate newquestions and new points of view concerning the operation of heat machines. Nevertheless, in atwo-level system an energy population swap follows immediately from (6) since only oneparameter is needed to describe the population transfer in a two-level system. For example, theEinstein rate equation for an interaction of two levels with thermal radiation describes such anenergy population swap interaction.

In appendix A, we show examples of two (partial) swap Hamiltonians. One generates adensity matrix swap and the other an energy population swap.

2.5. The adiabatic evolution step

Stroke A and C are termed ‘adiabatic’ as we require that the evolution will be diagonal in theenergy basis at the beginning and the end of the stroke. The coherences will remain zero and theenergy populations will remain as they are. Such a process does not have to be slow. Forexample, consider the following Hamiltonian that commutes with itself at all times:

∑==

H E t i i( ) . (13)i

N

iadiab

1

A density matrix that is diagonal in the energy basis | ⟩i will be invariant under this type ofHamiltonian regardless of how fast the energy levels are changing. In a two-level spin systemthis Hamiltonian will be σ=H B t( )z zadiab , where σz is the z Pauli matrix.

The population and coherence evolution can be obtained by more general time dependentHamiltonians that do not commute at different times ≠ ≠H t H t t[ ( ), ( )] 01 2 1 . Nevertheless thepossibility of an adiabatic transformation is guaranteed by the coherent control theorem[62, 63]. Alternatively it is possible to use a method known as ‘quantum driving’ or ‘shortcut toadiabaticity’ [64–69] to generate an evolution that preserves the energy-basis diagonal form ofthe density matrix at the end of the process.

In principle, we allow the energy levels to cross1. However in such a case it important toremember that the energy index is a level index and not some order index that indicates how thelevels are ordered. For example, in figure 1(b) < < <E E E Ec c c c,1 ,2 ,3 ,4 but at the hotbath < < <E E E Eh h h h,4 ,1 ,2 ,3.

3. The average populations in steady state operation

Next we explore the dynamics and the properties of the systemʼs (engine) reduced density ρs.We assume that during stroke B a single particle interacts with the engine with probability R.The description can easily be generalized to include the possibility of more than one collisionduring each thermal stroke. The probability R appears naturally in a collisions model as there is

1 This will not automatically generate strong non-adiabatic effects (e.g. if (13) drives the system).

7

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

no certainty that a particle from the a bath particle will be available to interact with the engine.For simplicity we assume that the bath parameters are identical for both baths: they have same Rand the same swap parameter x. A generalization to different x and R for different baths isstraightforward using the same methods.

Even though for some applications transient behavior may be of interest, here we focus onthe steady state operation. The average engine population at stage C, ρs

C, is related to that ofstage A, ρs

A, via:

ρ ρ ρ ρ= − + − +⎡⎣ ⎤⎦R R x x(1 ) (1 ) . (14)sC

sA

sA

h

The first term describes a no-collision event and the other describes a collision with a swapparameter x. Equation (14) simplifies to:

ρ ρ ρ= − +xR xR(1 ) . (15)eC

eA

h

Here x and R are inseparable since both of them have the same effect on the average population.In the same manner we can write the equation for ρs

A:

ρ ρ ρ= − +xR xR(1 ) . (16)sA

sC

c

By combining (15) and (16) we get:

ρρ ρ ρ

=+ −

−xR

xR2, (17)s

C h c c

ρρ ρ ρ

=+ −

−xR

xR2. (18)s

A h c h

Note that a combination of Gibbs states is not a thermal state if there are more than two levels.Hence, as expected in a finite time thermodynamic framework, the multilevel swap engine isnever in a Gibbs state for <xR 1. The expectation value of the population change is:

ρ ρ ρ ρ ρ= − =−

−→ xR

xRd

2( ). (19)s

A CsC

sA

h c

Inspection of (17) and (18) shows that even if the initial density matrix of the engine has somecoherences they have no impact on the energy diagonal steady state. Thus for all energyobservables computed locally (i.e., with the reduced density matrices of the baths or the engine)it is sufficient to consider the population vector ρ=p diag( )i i where = ′ ′ ′ ′ ′ ′i s c h, , . Hence, inthis notation:

=−

−→ xR

xRp p pd

2( ). (20)s

A Ch c

In general, the population change under a swap operation in steady state is proportional to−p ph c even when x and R are not the same for both baths. This result will have a large impact

later on. We point out that if the baths are only coupled to some of the system levels as in [6],then the result is different.

When xR = 1 in a single collision, or <xR 1 with a vast number of collisions a completethermalization of the engineʼs population takes place. Therefore, many of the results in the limitxR = 1 apply to a much more general setup than the swap collision used above. They apply

8

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

wherever the engine is connected long enough to effectively reach a Gibbs state (and not someother state).

For work and efficiency investigations, only the populations in the energy basis areimportant. Hence, it is not necessary that the whole density matrix is swapped as in (7)–(9),rather it is sufficient that an energy population swap takes place (see (10)–(12)).

3.1. Quantum versus classical swap and entropy generation

In equation (20) x and R are lumped together in a product form xR. Yet, they are not physicallyequivalent. R controls how deterministic the system is, and x determines the interaction strengthand the ‘quantumness’ (coherences and entanglement before the partial trace). Note that thequantum behavior is not monotonically increasing with x, since the system is classical whenx = 1 (bath and engine particles switch places).

In the quantum case, where R = 1 and <x 1, x can generate coherences and entanglementin the joint density matrix of two interacting particles. The increase in the sum of entropies ofthe reduced density matrices is the result of ignoring entanglement and classical correlations.

In contrast, in the classical case where x = 1, and <R 1, the particles are either fullyswapped or left as they are. Therefore entanglement cannot be produced from the initial productstate. Here, the sum of the entropies of the individual particles also increases since theinformation if a collision took place or not is discarded.

In both cases the increase of the bath particle entropy is encapsulated in the mutualinformation of the colliding particles. In the quantum case, the mutual information contains acontribution from entanglement. The separation to quantum and classical correlations can bestudied using the quantum discord tool [70]. This, however, is outside the scope of this work.Furthermore in our model the observables we studied depend only on xR so they can equally beobtained from a classical or a quantum realization. However, we do not expect this to hold forobservables that are not functions of the steady state population.

4. Thermodynamic properties

4.1. First law

In this model the expectation value of the engineʼs energy is:

= ·( )U

t tp E

dd

dd

(21)s

= · + ·

−

t tE p p E

dd

dd

. (22)s

t

s

tdd

Heat dd

Work

For the identification of work and heat see, for example [71, 72]. In short, work is associatedwith internal energy change when the population is fixed in time, and heat is a change in energywhen the Hamiltonian is fixed in time. On average in a complete cycle =U Uinitial final wehave:

9

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

Δ = = − + · −

+ · − + · −

( )( )

( )

( )

U p E E E p p

p E E E p p

0

, (23)

sA

h c h sC

sA

sC

c h c sA

sC

where the first term on the right-hand side corresponds to stroke A, the second term to stroke B,and so forth. Regrouping we identify:

= · →Q E pd , (24)h h sA C

= − · →Q E pd , (25)c c sA C

= · −→ ( )W p E Ed , (26)sA C

h c

where we used the ‘positive heat in, positive work out’ sign convention. Note that thesequantities are invariant to any constant shift of the levels in one or two of the baths:

→ + → +E E G E E G,h i h i h c i c i c, , , , where Gc and Gh are some constants. The probabilitiesthemselves are invariant to such transformation by virtue of the form of the Gibbs state. In theheat expressions, the energy is clearly not invariant to such a shift but the extra term cancels outwhen summed over the probability difference. Equations (24)–(26) lead to the averaged form ofthe first law:

+ =Q Q W . (27)h c

Note that the energy at the end of the cycle is equal to the energy at the beginning of the cycleonly on average. In a specific cycle it is, in general, not zero. Using (20) and (26) we obtain:

=−

− · −( )WxR

xRp p E E

2( ) . (28)h c h c

As mentioned before, the limit xR = 1, (28) holds for any interaction that leads to acomplete thermalization (or very close to it) and not just for a swap interaction. The modes ofoperation of a two-level swap machine are shown in figure 2. In a general multilevel system it isnot straightforward to write analytically the condition for engine or refrigerator operation usingthe energy levels and the temperature due to the exponential dependence of the probabilities onthe energy levels. Yet, in some temperature regimes this is considerably simpler as discussed insection 7.

4.2. The second law and the Clausius number

In cyclic processes in classical thermodynamics the second law can be expressed in terms of theClausius inequality ∮ ⩾δ 0Q

T. In our case we calculate:

=˜

+˜

Q

T

Q

T, (29)

h

h

c

c

where, as before, the ⟨ · ⟩ brackets denotes the average value in steady state and the tildesignifies a heat flow to the bath and not to the system (the sign is opposite). Hereafter we shallrefer to as the Clausius number. Strictly speaking, this is not the classical Clausius inequality.The swap collision process is not an isotherm at all. For a multilevel system a temperaturecannot even be assigned to the bath or engine particles since they are not in a Gibbs state after

10

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

(or during) the collision. In this section we find that ⩾ 0 holds in steady state for anymultilevel swap engine and that it has an information theoretic interpretation. Furthermore, it isshown that ⩾ 0 belongs to a family of more general inequalities. It is easy to verify that

⩾ 0 entails within the second law. For =T Tc h we get ⩽W 0. That is, in steady state, nowork can be extracted from a single bath. Alternatively, when calculating the efficiency,η = − |⟨ ⟩ ⟨ ⟩|Q Q1 c h , ⩾ 0 ensures that the efficiency is smaller than the Carnot efficiency(see section 6.2).

Using the expressions for heat we want to show that for swap heat machines:

∑= − ⩾=

→⎛⎝⎜

⎞⎠⎟ p

E

T

E

Td 0. (30)

i

N

iA C h i

h

c i

c1

, ,

In a specific cycle this does not have to be true. Our aim is to show that it holds on averagewhen the system is in steady state. To obtain the average we use (20) and get:

∑=−

− −⎛⎝⎜

⎞⎠⎟ xR

xRp p

E

T

E

T2( ) (31)

ic h

h i

h

c i

c

, ,

∑=−

+xR

xRp

p

pp

p

p2ln ln , (32)

ic i

c i

h ih i

h i

c i,

,

,,

,

,

which leads to the following result for swap engines:

=−

xR

xRJ p p

2( , ), (33)c h

Figure 2. A typical work per cycle curve (blue-solid) of a two-level partial swap heatmachine as a function of the cold bath energy gap, ΔEc, when ΔEh is held fixed. Belowthe Carnot point (left vertical line, Δ Δ =E E T Tc h c h) the device acts as a refrigeratorand above it until Δ Δ=E Ec h it performs as an engine. For Δ Δ>E Ec h the deviceperforms as a heater as it takes work to make the cold bath hotter. The dashed-red curveshows the total entropy of the baths and system.

11

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

= ∣ + ∣( ) ( )J D Dp p p p , (34)c h h cKL KL

where J is the Jeffreys divergence and | = ∑D pp q( ) lni ip

qKLi

iis the Kullback–Leibler

divergence or the relative entropy. A similar result is known for processes that start or end in athermal state [71, 73]. Since | ⩾D p q( ) 0KL for any two probability vectors (also known asKleinʼs inequality), ⩾ 0 follows naturally as an information theory inequality.

4.3. Generalized Clausius inequality

The Clausius number inequality is a special case of a more general inequality that holds forswap heat machines. We find that the more general inequality is:

∑= − ⩾−=

→−⎛

⎝⎜⎞⎠⎟ p

E

T

E

Td 0. (35)m

i

N

iA C h i

h

c i

c

m

2 1

1

, ,2 1

Clearly, if this equality holds, it holds for any odd analytic and monotonically increasingfunction of −E

T

E

T

h i

h

c i

c

, , as well. While ≡ 1 can be understood in thermodynamics terms ofheat, temperature and entropy, for >m 1 higher energy powers are involved and there is nostraightforward thermodynamic interpretation.

The proof of (35) is straightforward. Denoting ε = E Tb i b i b, , and ε ε= −Di h i c i, , we getthat inequality (35) is equivalent to:

∑=∑ ∑

−ε εε ε

−−

−′

−− − − − −

′ ( )D

e ee e e . (36)m

xR

xR

k k ij

D Di

m2 1

(2 ) 2 1c k h k

c i c j j i

, ,

, ,

Next we write = +− − − m m m2 11

2 2 11

2 2 1, exchange the indices names in the second term andobtain:

∑ ∑

∑

= −

× − −

ε ε

ε ε

− −

′

−

− − − − − −

′

( )( )

N

xR

xR

D D

12

2e e

e e e . (37)

m b

k k

ij

D Di

mj

m

2 1

2 1 2 1

c k h k

c i c j j i

, ,

, ,

The term −− −D Dim

jm2 1 2 1 has the same sign as −D Di j and the same sign as −− −(e e )D Dj i ,

hence the product of the two differences that appears in (37) is always positive. The rest of themultipliers are positive and symmetric under ↔i j and therefore ⩾− 0m2 1 .

4.4. Clausius dominated level

An immediate consequence of this generalized Clausius number inequality follows fromconsidering → ∞m . In this case, the term with the largest | − |E T E T( ) ( )h i h c i c, , becomesenormously larger than all the other terms. Therefore, this single term in ∞ must be positive. Ifthis term is positive, it is positive also in 1 and therefore the Clausius dominated level imax

defined by:

12

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

− ⩾ −E

T

E

T

E

T

E

T, (38)

h i

h

c i

c

h i

h

c i

c

, , , ,max max

satisfy:

= −→⎛⎝⎜

⎞⎠⎟p

E

T

E

Tsign d sign , (39)i

A C h i

h

c i

c

, ,

max

max max

where we assume that −E T E T( ) ( )h i h c i c, , has a single maximum and that ⟨ ⟩ ≠→pd 0iA Cmax

.Equation (39) allows us to immediately determine the direction of heat flow into the baths forthis level. Of course, this can be done explicitly by evaluating the probabilities numerically.This equality is not trivial since the Clausius dominated level is not necessarily the largestelement in the Clausius number sum (30).

4.5. Local and non-local quantities

We call a quantity ‘local scalar’ if the scalar quantity can be written in terms of the single-bathparameters Eb and Tb (i.e. f TE( , )b b ). The mean energy ·p Eb b of the bath particle, its purity

·p pb b, its entropy, its energy variance, its free energy and so on, are all examples of localscalars. A local function is a function of local scalars.

Many other quantities of prime importance cannot be written in this form: the indexcoincidence [74] ·p pc h, the fidelity of the two baths ∑ p pi c i h i, , , the Jeffreys andKullback–Leibler divergence [75], the Wootters distance [76] etc. Hence these scalars are ’non-local scalars’. More importantly heat, work and efficiency are non-local scalars.

In our analysis, thermodynamics amounts to finding relations between non-local scalars ofinterest (e.g. work and efficiency) and local scalars (temperature entropy, etc).

4.6. Purity reduction

Let us define the purity of the bath after the collision by the purity of the reduced densityformed by tracing out the engine and the other bath. Although the purity does not have all theappealing properties of the von Neumann entropy, it provides a simple and convenient measureof impurity. Furthermore, in contrast to entropy, it is not sensitive to whether we keep track ofthe particlesʼ position or not (i.e. there is no mixing entropy issue). In addition the purity willnaturally emerge in the derivation of bound on maximal work and efficiency.

The purity change in one cycle in one bath is:

Δ = + − = + · p p p p p pd d 2 d , (40)b b b b b b b2 2 2

where the absolute value of a vectors refers to the standard L2 norm = ∑ ≡= yy yiN

i12 2 .

Using = −p pd dc h for steady state operation we obtain:

Δ Δ+ = −⎜ ⎟⎛⎝

⎞⎠ xR

xRp2

1d (41)h c h

2

= − −−

−xR xR

xRp p

(1 )

(2 ). (42)h c2

2

In a refrigerator the cold bath purity increases while the hot bath becomes more mixed. Yet, thechange in the hot bath is larger so the purity of the whole system decreases. For <xR 1 the sum

13

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

of individual purities of the whole system (engine+baths) becomes increasingly more mixed inall modes of operations.

In the spirit of section 4.5 we want to express the purity reduction using local functions.This can be achieved by applying the inverse triangular inequality:

Δ Δ+ ⩾ −−

− xR xR

xRp p

(1 )

(2 )(43)h c h c2

2

= −−

− xR xR

xR

(1 )

(2 ). (44)h c2

2

Interestingly, in contrast to the Clausius inequality, we did not assume a thermal distribution. Infact, in steady state this device will decrease the total purity for any bath population that isdiagonal in the energy basis. While the Clausius number is not well defined (there is no notionof temperature for non thermal baths) the purity decrease still holds.

Finally, we note that using the Jensen inequality for the ln function it is straightforward toshow that the purity is related to the entropy through:

⩾ − e , (45)S

where = − ∑S p plogi i i is the standard entropy function. This inequality can be interpreted inthe following way: the Chebyshev sum inequality yields ⩾ N1 where the equality holds foruniform distributions. eS is the Shanonʼs effective number of degrees of freedom. The right-hand side of (45) is also equal N1 for uniform distribution. Hence (45) expresses the relationbetween two measures that count the degrees of freedom in the system.

5. Index of coincidence necessary conditions for engines and refrigerators

The engine regime is defined by the condition

= − · + · − · − · ⩾( )W xR xR p E p E p E p E(2 ) 0.h h c c c h h c

Using the mean bath energy = ·E p Eb b b and the free energy defined through = − −p eb

Eb i FbTb

,

we get:

∑= + + − + −( ) ( )W E E p T p F p T p Fln ln . (46)h c

ic i h h i h h i c c i c, , , ,

After rearranging and using the Jensen inequality to get

∑ ∑· ⩾ ⃗ · ⃗ ⩾p p p p p pp pln ln , ln lnc h i c i h i c h i h i c i, , , ,

we get that the work satisfies the inequality:

⩽ + + + ·( )W T S T S T T p pln , (47)c c h h h c h c

where we used: = −T S E Fb b b b. If the right-hand side is smaller than zero so will be W .Thus, demanding that the right-hand side of (47) is positive we get a necessary (but notsufficient) condition for the heat machine to perform as an engine:

⩾ −++ e , (48)ch

T S T ST T

c c h h

c h

14

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

where ch is the index of coincidence ·p pc h [74]. While the purity describes the probabilitythat the same result will appear when measuring the energy of two particles in the same bath,ch describes the probability that particles from different baths will be in the same level. ch isoften used in cryptography and code ciphering. Note, that it does not imply a correlation butsimply the likelihood of identical events in both baths.

Equation (48) also imposes a restriction on the individual purities since:

⩽ · ⩽ ⩽+−

++

p pe2

, (49)T S T S

T T c h c hc hc c h h

c h

where we used the Cauchy–Schwarz inequality to go from the second step to the third. The laststage before the end of (49) can be expressed using the second order Rényi entropy [77]

= − S lnR b b,2 :

++

⩾+T S T S

T T

S S

2. (50)c c h h

c h

R c R h, ,2 2

Now we wish to find the analogue condition for the refrigerator regime. Cooling occurs whenthe cold bath gives away heat to the system2:

− · >Ep p( ) 0. (51)c h c

Repeating the same procedure as before we get a necessary refrigerator condition:

⩾ − e . (52)chSc

In contrast to other measures like fidelity, ch has a very simple statistical interpretation whichalso makes it easier to measure it in practice. ∑ p pi c i h i, , is the probability that two particleschosen from different baths will be in the same state (like getting the same result with twodifferent unbalanced dices). To evaluate it, there is no need to keep track of the exact result andthen to estimate the probabilities pc i, and ph i, through their frequencies. One only needs to keeptrack of if the results are the same or not. Thus, ch corresponds to a binary random variable (acoin flip). Like the purity, ch also satisfies ⩾ N1ch (if the levels do not cross) but this is truefor any two monotonic distributions and in contrast to (48) and (52) it does not give anyindication of the heat machineʼs functionality.

6. Upper bounds on work and efficiency

6.1. Bounds on the maximal work production

The first work upper bound can be obtained from (47) where the inequality· ⩽ p pln lnc h c h

1

2is used to bring it to the local form:

⩽−

+ ++⎡

⎣⎢⎤⎦⎥ W

xR

xRT S T S

T T

2 2ln . (53)h h c c

h cc h

2 For the refrigerator regime the work is not a good indicator since it is possible to apply work to the systemwithout cooling the cold bath (e.g. the rightmost regime in figure 2).

15

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

A different bound can be obtained by writing the work in a different form:

=−

− −

− ∣ − ∣

⎡⎣⎤⎦

( )( )

( ) ( )

WxR

xRT T S S

T D T Dp p p p

2

. (54)

h c h c

c h c h c hKL KL

The last two terms are always negative (including the minus sign). Therefore the first term mustbe positive for the machine to perform as an engine. ⩾S Sh c is expected for engines as enginetransfer entropy from the hot bath to the cold bath but (54) quantifies how large the entropydifference must be:

− ⩾∣ + ∣

−( ) ( )

S ST D T D

T T

p p p p. (55)h c

c h c h c h

h c

KL KL

The work expression (54) contains within the non-local quantity DKL. To obtain a local upperbound we use the inequality:

∑∣ ⩾ −

⩾ + −

= − ( )( )

( )

( )D p p p p

p p p p

1212

2

12

, (56)

c h c i h i

c h c h

c h

KL , ,

2

2 2

2

and get:

⩽−

− − −+

−⎡⎣⎢

⎤⎦⎥ ( )( )( )W

xR

xRT T S S

T T

2 2. (57)h c h c

c hc h

2

Different approaches and different inequalities may lead to work upper bounds that can performbetter in certain regimes. In addition, further reasonable restrictions on the system can also leadto better results. In appendix C we derive another work bound under some assumptions aboutthe energy levelsʼ structure.

6.2. Upper bounds on the efficiency

In the two-level case the efficiency is simply given by − −−

1 E E

E E

c c

h h2 1

2 1and the population change

cancels out. In the multilevel engine the efficiency is:

η = −··

p E

p E1

d

d. (58)

c

h

Using the Clausius number defined in section 4.2:

η = − −·

T

T

T

p E1

d. (59)c

h

c

h

Notice that this result is still exact. Since ⩾ 0 we get that the swap engine efficiencyis always smaller than the Carnot efficiency. To get a tighter upper bound we use

16

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

again the inequality | ⩾ −D p p p p( )c h c hKL1

2

2and Cauchy–Schwarz in the denominator and

obtain:

η ⩽ − − −T

T

Tp p1 , (60)c

h

c

hc h

where we used h to denote the centered energy vector:

∑= −=

EN

E1

. (61)h i h i

k

N

h k, ,

1

,

The replacement → Eh h is highly important. It keeps the bound invariant to energy shifts andat the same time makes the bound tighter. Equation (60) can be written as

η ⩽ − − + − T

T

T1 2 . (62)c

h

c

hc h ch

In order to obtain a local-quantities bound we use the inequality ⩽ ch c h and obtain aweaker yet local upper bound:

η ⩽ − − − T

T

T1 . (63)c

h

c

hc h

This inequality is often very close to the Carnot efficiency. Yet, it still shows that the bathsʼpurity imposes some limitations on the efficiency.

A different upper bound can be written down in term of the Wootters statistical distance[76] between the hot and cold probability distribution:

∑==

⎛⎝⎜⎜

⎞⎠⎟⎟ p parccos . (64)w

i

N

c i h i1

, ,

From [73] it follows that the Jeffreys divergence satisfies:

π⩾ J

16. (65)w2

2

The efficiency bound obtained from using (65) in (59) is not always smaller or larger than (60)and it uses the non-local quantity ∑ = p pi

Nc i h i1 , , (fidelity). Yet, it introduces another relation

between the difference in the statistics of the baths and the efficiency.

7. The ultra-hot baths and the quasi-static regime

7.1. The ultra-hot baths regime

For systems with small energy gaps (e.g. magnetically induced splitting or translationalmotion in a trap), the levels are highly excited even for low bath temperatures. The condition

≪E T E T, 1h h c c defines a high temperature limit for the specific system (a morerigorous condition will be given later). In this section we study the multilevel heatengine operation when the baths are so hot that we consider only the first order correction in

17

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

T1 b to the → ∞Tb limit where →p N1b i, . The ultra-hot regime is defined by the smallparameters:

Δ Δ≪

E

T

E

T, 1, (66)c

c

h

h

where Δ = −E E Eb b b,max ,min is the gap of the bath. First order expansion inβ β= =T T{ 1 , 1 }c c h h yields:

β β β β=−

+ · − −‐ ⎡⎣ ⎤⎦ ( )WxR

xR N21

. (67)c h c h c c h hultra hot 2 2

This time the centered energy form b emerges naturally from the free energy normalizationfactor in the ultra-hot limit. >W 0 yields the ultra-hot necessary and sufficient enginecondition:

β ββ β

· >++

. (68)c h

c c h h

c h

2 2

Applying Cauchy–Schwarz we get a necessary condition for ultra-hot swap engines:

< <

T

T1 . (69)

h

c

h

c

Work per cycle optimization

Next we want to take advantage of the simple energy dependence in the work expression (67)and optimize the work output under some restrictions. First, let us assume that the norms of theenergies c and h are fixed. Under these two constraints the last two terms in (67) are fixed.To maximize the first term, and consequently the work, c must be parallel to h :

= , (70)h i c i, ,

⩽ ⩽ T

T1 , (71)h

c

where is the compression ratio and (71) follows from (69). For ‘uniform compression’ (70)the device works as a refrigerator when < 1, and as a heater when ⩾ T Th c. In the regime(71) it performs as an engine. The uniform compression can be studied for any temperature butin the ultra-hot regime it is found that the uniform compression maximizes the work when theenergy norms are fixed. Now we remove the restriction that c is fixed and use (70) in (67).Imposing ∂ =W 0 we get:

=+

⩽ ( )

T

T T2, (72)W

h

h c, 1

2

hmax

where the h subscript signifies the = consth optimization constraint. This result is abit surprising. If the hot levels are fixed we find that there is no point in compressing them bymore than a factor of two (or max, h

to be exact) to reach the maximal output work.The efficiency is:

18

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

η η= − =−

= ⩽‐ T T

T1 1

212

12

, (73)Wh c

hcmax,

ultra hot,h hmax

where ηc is the Carnot efficiency. The optimal compression and the efficiency at maximumwork critically depends on the constraint we impose. If we impose that the cold bath energynorm is fixed and optimize over the hot bath levels we obtain:

=+

( )T T

T, (74)W

h c

c,

1

2cmax

ηη

η=

−+

=−

‐

T T

T T 2. (75)W

h c

h c

c

c,

ultra hotcmax

Interestingly, the same expression for the efficiency was obtained in [78] for a classical engineoperating at maximum power.

In [79] both (73) and (74) were obtained for slow classical engines when the engine isclose to reaching equilibrium with the bath during the thermal strokes. Depending on whetherthe hot bath relaxation time of the cold bath overwhelmes the cold bath relaxation time or viseversa (73) and (74) are obtained. If the time scales are the same (the symmetric case) then theChambadal, Novikov, Curzon and Ahlborn (CNCA) efficiency bound [53, 80, 81]η = − T T1CA c h is obtained. In our system we can get CNCA result as well by imposingthe constraint = constc h (or equivalently fixing the geometric mean of the norms) thatlead to:

= T

T, (76)W

h

c, c hmax

η η= − = − −‐

T

T1 1 1 . (77)W

c

hc,

ultra hotc hmax

Despite the similarity of our final results (73), (74) and (76) to [79] the physical scenario iscompletely different. First, we optimize work per cycle and not power. Second, our result doesnot reach the Carnot efficiency if the cycle (the collisions) is slower. In our case, the deviationfrom Carnot is due to the levelsʼ structure and the deviation persists regardless of the degree ofthermalization. To see this one can take the swap parameter to be one where completethermalization takes place or take it to be tiny where the state is hardly affected by the bath.However the efficiency at maximal work is independent of the swap parameter and the level ofthermalization.

Once Wmaxis found then together with the constraint it can be used to get an expression for

the maximal work. Let us write it explicitly just for the = consth case in order to clarify apoint:

=−

−‐ ( )

WxR

xR N

T T

T T21

4. (78)

h c

c hhmax,

ultra hot2

22

h

At first sight, it may seem that theN

1 term suggests that the work becomes smaller whenthe number of levels increases. Yet, h

2 may also depend on N. For example if thelevels are degenerate so that there are N 2 replicas of two levels, we get that the work does notdepend on the number of replicas, as expected.

19

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

To conclude this section we evaluate the Clausius number in the ultra-hot limit and get:

∑

∑

=−

− −

=−

− − +

‐⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

xR

xR T T

E

T

E

T

xR

xR T T T T

2

2const . (79)

i

h i

h

c i

c

h i

h

c i

c

i

h i

h

c i

c

h i

h

c i

c

ultra hot , , , ,

, , , ,

The constant term can be dropped since it contributes zero to the total sum. After the constantterm is removed, all the terms in the Clausius sum are positive. This is not true for temperaturesbelow the ultra-hot bath regime. In this regime, the Clausius inequality holds since theprobability difference and the Clausius factor − T T( ) ( )h i h c i c, , have the exact same form.

7.2. Almost ‘quasi-static evolution’ at finite time

Consider the case where the swap parameter is sufficiently small so that after a collision thechange in a bath particle population, pd b i, , is small with respect to the original bath populationδ ≪p pb i b i, , . At the end of appendix A it is shown that the energy basis diagonal form of thedensity matrices is conserved in an energy population swap (this trivially holds in a densityswap interaction). Consequently, it is possible to use just pb and δpb to calculate the vonNeumann entropy change. To first order in δpb i, the change in the bath particleʼs entropy isgiven by:

δ δ δΔ δ

= ⃗ + ⃗ − ⃗ = =˜

( ) ( )S S p p S p pE

T

Q

T. (80)b b b b b

b

b

b

b

Note that this equation holds only for the bath particles and for tiny deviations from theGibbs state. The engine is not in a thermal state at all, and cannot be assigned a temperature.However, in a complete cycle, on average, the engine returns to its initial state so the enginedoes not contribute to the average entropy production of the total system. Therefore the totalincrease in both baths and the system satisfies:

δ δ+ ⩾S S 0, (81)h c

by virtue of the Clausius inequality for swap heat machines proven earlier.Assuming that on average there are n cold collisions and n hot collision, the work can be

expressed in terms of the entropy changes:

δ δ= +W nT S nT S . (82)h h c c

Although (80) seems like a plausible property for a bath, it is not mandatory and in ourmodel it emerges only in the weak coupling limit (small population change per collision).

Even though we assumed the change in a single collision is small it does not mean that theheat exchange in strokes B or C must be small in our model. If we allow multiple collisions, it ispossible to have large changes while still satisfying Δ Δ δ˜ = =Q T S T n Sb b b b b.

In our entropy considerations we have not included the mixing entropy so it is not claimedthat (80) is the change in the entropy of the system but just the part of the entropy change that isresponsible for the heat exchange.

20

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

8. Conclusion

We have presented an analysis of a multilevel heat machine that is driven by a sequential partialswap interaction with a hot and cold bath. In the limit of the complete swap our results describeany ‘slow’ four-stroke Otto cycle where the interaction with the bath (not necessarily a swapinteraction) has sufficient time to bring the engine very close to thermal equilibrium. Wederived necessary conditions for the engine and refrigerator regimes and produced refined upperbounds on the work and efficiency using the relative entropy and Jeffreys divergence of thebaths. Our analysis emphasizes the difference between ‘local’ quantities that are evaluated usingthe properties of a single bath, and ‘non-local’ quantities that require both baths to be evaluated.Bounds on non-local quantities of interest such as work, heat and efficiency were expressed interms of simpler local quantities such as the purity and entropy of the baths.

The equivalent of Clausius inequality in this system was generalized to higher orderenergy moments. We identified a quasi-static regime in finite time evolution and an ultra-hotregime where stronger statements can be made and work optimization can be carried outanalytically.

While our findings are valid for any multilevel engine and collisions that can bring thesystem to a Gibbs state, it is interesting to consider other bath models like the ones used incontinuous engines where different levels interact with different baths or where thethermalization rate is different for different levels.

Acknowledgments

This work was supported by the Israel Science Foundation. Part of this work was supported bythe COST Action MP1209 ’Thermodynamics in the quantum regime’.

Appendix A. The two-level swap Hamiltonian

The partial swap unitary is given by:

=∑ϕ σ σ− ⊗

U e , (A.1)i 12

i

i i

where σi are the Pauli matrices and ϕ is the swap angle. ϕ π= 2 corresponds to a completeswap. The reduced density matrix of particle 1 after applying U is:

ρ ρ ρ= ⊗′ †( )U Utr . (A.2)1 2 1 2

One can verify that the density swap rule is satisfied:

ρ ϕρ ϕρ= +′ cos sin . (A.3)12

12

2

Since the Hamiltonian in the exponent of U is written in terms of the Pauli matrices it is notstraightforward to generalize it to a multilevel swap operation. For energy population swap, thegeneralization is simpler when two-particle states are considered. A complete swap operationyields: | ⟩ = | ⟩U i j U j i, ,cs cs . The states | ⟩i i, are invariant under this operation. The completeswap unitary in the two-particle state can be written as:

21

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

ðA:4Þ

The column to the left shows how the states are ordered in the two-particle density matrix.In this form the unitary has a clear block diagonal structure. In the space of each non-identicalstate | ⟩ | ⟩ij ji{ , } a simple spin flip takes place:

+ → +a ij b ji a ji b ij . (A.5)

Replacing the spin flip by a more general rotation in this Hilbert subspace, it is easy to deducethe partial swap time-independent Hamiltonian:

∑ϕ=H ij ji . (A.6)ij

ij

The ϕii terms just contribute a phase to the invariant states. A complete swap takes placewhen ϕ π= ±≠ 2i j i, .

In general, there is no particular reason why the rotation rate, ϕij should be the same for allpairs of states. However, when it is the same, ϕ ϕ=≠i j can show that the energy populationswap rule follows:

ρ ρ ϕ ϕ= ⊗ = +′ †( )( )p U U p pdiag tr cos sin . (A.7)1 2 1 22

12

2

This swap model is specially designed for the energy basis. Consequently, for partial swap, ingeneral, it does not satisfy the ‘density swap’ rule but the energy basis probability rule. Yet, aproper choice of ϕii leads to the density swap Hamiltonian that appears in the exponent of (A.1).We conclude by noting the important fact that the energy population Hamiltonian preserves theenergy diagonal form after the partial trace. That is, if the input state is ρ ρ⊗1 2 where ρ ρ,1 2 arediagonal in the energy basis then ρ ρ ρ= ⊗′ †U Utr ( )1(2) 2(1) 1 2 will be diagonal in the energy basisas well. For simplicity we illustrate this for the ϕ ϕ=≠i j case. The density matrix after thecollision and before the trace is:

∑ρ ρ ϕρ ρ ϕρ ρ⊗ = ⊗ + ⊗ +†

≠

U U f ij jicos sin , (A.8)i j

ij1 22

1 22

2 1

where fij are some complex coefficients. The third term vanishes when taking partial trace oneither particle. Due to the block structure (A.4) this holds even when ϕ ≠i j depends on i and j.

Appendix B. Markovian swap formulation

As shown in section 3, in the steady state all coherences of the density matrix vanish. Thus onlythe diagonal elements are important and the Markov chain formalism can be used to describethe evolution of the diagonal elements (probabilities). In this appendix it will be more

22

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

convenient to use Diracʼs ‘Bra–Ket’ notation for the probability vectors in order to distinguishbetween right vectors and left vectors. However the normalization is still the regular probabilitynormalization.

The probabilities before and after the collision with the hot bath satisfy:

=′p K p , (B.1)s h s

= ˜∣ ⟩ + − ˜ ×( )K x p x I1, 1, 1 .. 1 , (B.2)h T N Nh

where x̃ is an effective swap parameter that may contain a ‘classical’ contribution from thecollision probability R and a quantum contribution from a quantum partial swap

ϕ π< <x or( 1 2). ×IN N is the identity operator. One can verify that the above equationslead to the population swap rule ′ = − ˜ + ˜p x p xp(1 )s s h. A Markov chain with a single steadystate vector is always associated with a left eigenvector of the form ⟨ |1, 1, 1 .. . Thus, | ⟩pTh

is aright eigenvector of Kh and it has a unity eigenvalue: | ⟩ = | ⟩K p ph T Th h

.The engine operation over one cycle starting form stroke A is given by:

=

= ˜ + ˜ − ˜ + ˜ − ˜

+ − ˜

→

×

⎡⎣ ⎤⎦( ) ( )

( )

K K K

x p x x p x x p

x I

1 1 1, 1, 1 ..

1 . (B.3)

A A c h

T T T

N N

2

2

c c h

This operator has a clear interpretation. The ˜ | ⟩x pT2

cterm describes two complete swaps that

occur with probability x̃2. Since both swap events are complete, the population is determined bythe last swap with the cold bath. The ˜ − ˜x x(1 ) terms describe the probability for a singlecomplete swap event, and the − x̃(1 )2 term describes zero swap events.

Clearly the invariant steady state is − + | ⟩ + − | ⟩x x x p x x p( (1 ) ) (1 )T T2

c h. After normal-

ization we obtain the steady state eigenvector:

=∣ ⟩ + ∣ ⟩ − ˜∣ ⟩

− ˜p

p p x p

x2. (B.4)e

A T T Tc h h

Repeating this for a cycle starting from stroke C and using =→K K KC C c h one gets:

=∣ ⟩ + ∣ ⟩ − ˜∣ ⟩

− ˜p

p p x p

x2. (B.5)e

C T T Tc h c

Both (B.4) and (B.5) agree with the steady state population obtained in (3), whenreplacing ˜ →x xR

Appendix C. Alternative work upper bound

The expression for the work is a standard inner product over the real vectors. By using theCauchy–Schwarz inequality we get:

⩽−

− −WxR

xRp p E E

2. (C.1)c h c h

At first, it seems, that this separation between statistics and energy is not very useful. If all ⃗pcand ⃗ph have to be known, then there is not much difference from calculating the exact value of

23

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

the work. However, −p pc h , can be expressed in term of simple scalars quantities thatcharacterize the baths.

∣ − ∣ = ∣ ∣ + ∣ ∣ − · = + − p p p p p p2 2 . (C.2)c h c h c h c h ch2 2 2

The upper bound (C.1) holds for any diagonal distribution and even if the distribution isnot known exactly it is enough to measure ,c h and ch to evaluate ∣ − ∣p pc h . Since pc and phare ordered if there are no level crossings, we can use Chebyshevʼs sum inequality and get

· ⩾ Np p 1c h so that:

⩽−

+ − − WxR

xRE E

22 (C.3)c h ch c h

⩽−

+ − − xR

xR NE E

22

. (C.4)c h c h

Note that: + − ⩽ − N N N(2 ) 2 ( 1)c h . If all the levels are compressed (notnecessarily by the same factor) so that:

= E E1

(C.5)c ii

h i, ,

⩾ 1, (C.6)i

in this case, we can use − ⩽ − + = ∣ − ∣ ∀ >a b a b a b a b ab( ) 02 2 2 and get thefollowing local form:

⩽−

+ − −⩾ WxR

xR NE E

22

(C.7)c h h c1i

⩽−

+ − − xR

xR NE E

22

. (C.8)c h h c2 2

We cannot use the centered level here, as it may violate (C.5). The advantage of this bound isthat it separates the statistical properties of the baths from the energy structure of the baths andthat all the quantities used are local.

References

[1] Horodecki M and Oppenheim J 2013 Nat. Commun. 4 2059[2] Åberg J 2013 Nat. Commun. 4 1925[3] Dahlsten O C, Renner R, Rieper E and Vedral V 2011 New J. Phys. 13 053015[4] Kosloff R 2013 Entropy 15 2100[5] Gemmer J, Michel M and Mahler G 2009 Quantum Thermodynamics (Lecture Notes in Physics vol 784)

(Berlin: Springer)[6] Geusic J, Schulz-DuBios E and Scovil H 1959 Phys. Rev. Lett. 2 262[7] Kosloff R 1984 J. Chem. Phys. 80 1625–31[8] Geva E and Kosloff R 1994 Phys. Rev. E 49 3903[9] Abah O, Roßnagel J, Jacob G, Deffner S, Schmidt-Kaler F, Singer K and Lutz E 2012 Phys. Rev. Lett. 109

203006[10] Jahnke T, Birjukov J and Mahler G 2008 Ann. Phys. 17 88[11] Scully M O, Zubairy M S, Agarwal G S and Walther H 2003 Science 299 862

24

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

[12] Svidzinsky A A, Dorfman K E and Scully M O 2011 Phys. Rev. A 84 053818[13] Rahav S, Harbola U and Mukamel S 2012 Phys. Rev. A 86 043843[14] Harbola U, Rahav S and Mukamel S 2012 Europhys. Lett. 99 50005[15] Skrzypczyk P, Short A J and Popescu S 2013 arXiv: 1302.2811[16] Brunner N, Huber M, Linden N, Popescu S, Silva R and Skrzypczyk P 2014 Phys. Rev. E 89 032115[17] Dillenschneider R and Lutz E 2009 Europhys. Lett. 88 50003[18] Geva E and Kosloff R 1996 J. Chem. Phys. 104 7681[19] Palao J P, Kosloff R and Gordon J M 2001 Phys. Rev. E 64 056130[20] Kosloff R, Geva E and Gordon J M 2000 J. Appl. Phys. 87 8093[21] Levy A and Kosloff R 2012 Phys. Rev. Lett. 108 070604[22] Kolář M, Gelbwaser-Klimovsky D, Alicki R and Kurizki G 2012 Phys. Rev. Lett. 108 090601[23] Segal D and Nitzan A 2006 Phys. Rev. E 73 026109[24] Linden N, Popescu S and Skrzypczyk P 2010 Phys. Rev. Lett. 105 130401[25] Wu L, Segal D and Brumer P 2013 Sci. Rep. 3 1824[26] Kim S W, Sagawa T, de Liberato S and Ueda M 2011 Phys. Rev. Lett. 106 070401[27] Zhou Y and Segal D 2010 Phys. Rev. E 82 011120[28] Roßnagel J, Abah O, Schmidt-Kaler F, Singer K and Lutz E 2014 Phys. Rev. Lett. 112 030602[29] Abah O and Lutz E 2014 Europhys. Lett. 106 20001[30] Correa L A, Palao J P, Alonso D and Adesso G 2014 Sci. Rep. 4[31] Geva E and Kosloff R 1992 J. Chem. Phys. 96 3054[32] Feldmann T, Geva E, Kosloff R and Salamon P 1996 Am. J. Phys. 64 485[33] Henrich M J, Rempp F and Mahler G 2007 Eur. Phys. J. Spec. Top. 151 157[34] Skrzypczyk P, Brunner N L N and Popescu S 2011 J. Phys. A: Math. Gen. 44 492002[35] Rezek Y and Kosloff R 2006 New J. Phys. 8 83[36] Beretta G P 2012 Europhys. Lett. 99 20005[37] Quan H, Zhang P and Sun C 2005 Phys. Rev. E 72 056110[38] Lindblad G 1976 Commun. Math. Phys. 48 119[39] Gorini V, Kossakowski A and Sudarshan E 1976 J. Math. Phys. 17 821[40] Levy A, Alicki R and Kosloff R 2012 Phys. Rev. Lett. 109 248901[41] Levy A and Kosloff R 2014 Europhys. Lett. 107 20004[42] Scarani V, Ziman M, Štelmachovič P, Gisin N and Bužek V 2002 Phys. Rev. Lett. 88 097905[43] Ziman M, Štelmachovič P, Bužek V, Hillery M, Scarani V and Gisin N 2002 Phys. Rev. A 65 042105[44] Ziman M and Bužek V 2005 Phys. Rev. A 72 022110[45] Gennaro G, Benenti G and Palma G M 2008 Europhys. Lett. 82 20006[46] Gennaro G, Benenti G and Palma G M 2009 Phys. Rev. A 79 022105[47] Burgarth D and Giovannetti V 2007 Phys. Rev. A 76 062307[48] Rybár T, Filippov S N, Ziman M and Bužek V 2012 J. Phys. B: At. Mol. Opt. Phys. 45 154006[49] Bodor A, Diósi L, Kallus Z and Konrad T 2013 Phys. Rev. A 87 052113[50] Ziman M, Štelmachovič P and Bužek V 2005 Open Syst. Inform. Dyn. 12 81[51] Ziman M, Stelmachovic P and Buzek V 2003 J. Opt. B: Quantum Semiclass. Opt. 5 S439[52] Salamon P, Nulton J, Siragusa G, Andersen T R and Limon A 2001 Energy 26 307–19[53] Curzon F and Ahlborn B 1975 Am. J. Phys. 43 22[54] Andresen B 2011 Angew. Chem. Int. Edn Engl. 50 2690[55] Disói L, Feldmann T and Kosloff R 2006 Int. J. Quantum Inform. 4 99[56] Allahverdyan A E, Hovhannisyan K and Mahler G 2010 Phys. Rev. E 81 051129[57] Diaz de la Cruz J M and Martin-Delgado M A 2014 Phys. Rev. A 89 032327[58] Attal S and Joye A 2007 J. Stat. Phys. 126 1241[59] Horowitz J M and Parrondo J M 2013 New J. Phys. 15 085028[60] Nechita I and Pellegrini C 2009 Confluentes Math. 1 249

25

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff

[61] Feldmann T and Kosloff R 2000 Phys. Rev. E 61 4774[62] Huang G M, Tarn T J and Clark J W 1983 J. Math. Phys. 24 2608[63] Ramakrishna V, Salapaka M V, Dahleh M, Rabitz H and Peirce A 1995 Phys. Rev. A 51 960[64] Demirplak M and Rice S A 2003 J. Phys. Chem. A 107 9937[65] Demirplak M and Rice S A 2008 J. Chem. Phys. 129 154111[66] Berry M 2009 J. Phys. A: Math. Theor. 42 365303[67] Chen X, Torrontegui E, Stefanatos D, Li J S and Muga J 2011 Phys. Rev. A 84 043415[68] Deffner S, Jarzynski C and del Campo A 2011 Phys. Rev. X 4 021013[69] del Campo A, Goold J and Paternostro M 2013 arXiv: 1305.3223[70] Ollivier H and Zurek W H 2001 Phys. Rev. Lett. 88 017901[71] Anders J and Giovannetti V 2013 New J. Phys. 15 033022[72] Esposito M and Mukamel S 2006 Phys. Rev. E 73 046129[73] Deffner S and Lutz E 2010 Phys. Rev. Lett. 105 170402[74] Friedman W F 1987 The Index of Coincidence and Its Applications in Cryptanalysis (Walnut Creek, CA:

Aegean Park)[75] Vedral V 2002 Rev. Mod. Phys. 74 197[76] Wootters W K 1981 Phys. Rev. D 23 357[77] Müller-Lennert M, Dupuis F, Szehr O, Fehr S and Tomamichel M 2013 J. Math. Phys. 54 122203[78] Gelbwaser-Klimovsky D, Alicki R and Kurizki G 2013 Phys. Rev. E 87 012140[79] Esposito M, Kawai R, Lindenberg K and Van den Broeck C 2010 Phys. Rev. Lett. 105 150603[80] Chambadal P 1957 Les Centrales Nuclaires (Paris: Armand Colin)[81] Novikov I 1958 J. Nucl. Energy 7 125

26

New J. Phys. 16 (2014) 095003 R Uzdin and R Kosloff